Bulgarian Chemical Communicaions, Volume 48, Special Issue E (pp. 59-64) 016 Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions K. B. Hadjov, A. S. Aleksandrov, M. P. Milenova, V. A. Aleksandrova * Universiy of Chemical Technology and Meallurgy, 8 Kl.Ohridski Blvd., Sofia 1756, Bulgaria In his work, in analyical form, are derived insananeous sress-srain and vice-versa relaions using he Neo-Hookean and he Mooney-Rivlin models. These relaions are obained resolving algebraic equaions of hird and fourh degree respecively. The above menioned analyical soluions are incorporaed in he herediary inegral equaions of Volerra o predic he creep and relaxaion of such as maerials a large deformaions. One akes ino accoun he non-lineariy of he viscous behaviour in he presence of similariy in he isochrone sress-srain curves. Our heoreical resuls are compared wih experimenal daa for wo kinds of rubbers and demonsrae very good coincidence. Keywords: rubber, mechanical properies, rheology, large deformaions INTRODUCTION Rubbers are increasingly used in modern indusry [1-4]. Resinous maerials and rubbers are elasoviscous solids. They are very deformable and possess non-linear behaviour. Their creep is also non-linear according o he applied sresses. The las non-lineariy can be observed excluding he ime from he creep curves (he so-called isochrones). Thus, rubbers require idenificaion and descripion of wo differen kinds of nonlineariies. Here we examine wo composiions: he firs one is he Buadienniril rubber (BN) and he oher one - he Polyisoprene rubber (PI) [5]. In he firs rubber composiion is used rubber wih 40% acryloniril in he macromolecule [5]. In he second composiion is used polyisoprene rubber - an analogue of he naural rubber. The microsrucure concise of 1,4 cispolyisoprene wih conen of hese unis almos 98% [5]. Boh elasomeric composiions include fillers (cinders) wih a developed surface respecively 75 and 50 m²/ g. THEORETICAL Nonlinear elaso-viscosiy a small deformaions The herediary linear equaions can be generalized o accoun he nonlinear mechanical behaviour using he Rabonov-Rzanizin approach [6, 7]. This approach requires similariy in he sress - srain curves for differen momens ( ) [ 1 f ( ) K ( τ) dτ] E, (1) * To whom all correspondence should be sen: aleksandrovassucm@gmail.com 0 where ( ) is he srain, - he applied sress, f ( ) - he nonlineariy funcion, K ( - ) - he creep kernel, - he curren ime and 0. Concerning he creep kernel we can say he following. I is he resolving kernel of he relaxaion one. As kernels in he inegral equaions of Volerra like eq. (1) i is recommended o ake singular kernels which beer describe he enhanced creep and relaxaion rae a he beginning. In his work we assume he singular relaxaion kernel of Kolunov [7] e R( ) A, (a) whose resolving kernel (he creep kernel) looks like [7]. e n n K( ) A( ) / ( n ), (b) n1 The funcion of he nonlineariy f ( ) is deermined from he isochrone curves. This presenaion of he nonlineariy involves a coincidence of he iniial poins a he srain scale and requires a similariy of he isochrones. If a low sresses (up o he limi axial sress of he lineariy o ) he behaviour is linear, equaion (1) can be generalized as follows: n ( ) [ 1 ( / o) K( τ) dτ] E () 0 In equaion () n is he poency which can be deermined from he isochrones. Eq. () remains valid if o. In he opposie case he behaviour is linear and / o 1. Such a presenaion of he viscous nonlineariy is used in many pracical problems [8]. 016 Bulgarian Academy of Sciences, Union of Chemiss in Bulgaria 59
K. B. Hadjov e al.: Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions Nonlinear elasiciy a large deformaions Rubbers are very deformable and heir elasic behaviour canno be described by he Hooke s law. Here as elasic behaviour i is mean he insananeous elasiciy of equaion () given by he raio before he brackes. This expression should be changed o be able o accoun he large elasic deformaions of he rubbers. Here i is used he heory of he large deformaions described in [9]. The maerial is considered as isoropic and incompressible. To describe he mechanical behaviour of rubbers and oher resinous maerials as successful models are acceped he neo-hookean and he Mooney-Rivlin ones [9]. According o he above models he following hermodynamic poenials are inroduced G W ( 1 ), (4а) 1 W ( - )( 1 ) (4b) ( 1 1 ) 4 where G is he second module of Young, ζ and χ are experimenally deermined parameers based on he insananeous force-exension curves and i are he large relaive exensions in he main direcions i 1 i. (5a) Here i 1,, and i are he main Cauchy s srains (he classic ones in he case of small displacemens). In he case of incompressible maerials 1 1 and by racion i follows 1/ 1,. (5b) Therefore, o he poenials (4a) and (4b) follows G -1/ W ( ), (6а) 1 ( - )( / ) (1/ W ) (6b) 4 In he heory of large deformaions [9] he axial ensile force is given by he expression dw Q( ) So, (7) d where S o is he iniial secion of he sample associaed wih he curren secion S( ) as So S( ). (8) So, based on equaions (6a), (6b) and (7) o he axial ensile force of highly deformable maerials are obained he following expressions (depending on he relaive exension) concerning boh models respecively 1 Q( ) SoG, (9a) 1 1 Q( ) SoG ( ( 1)) (9b) Relaions (9) mus be expressed hrough he deformaions of Cauchy and replaced in () in order o ake ino accoun he large deformaions. This can be done as follows. Based on eq.(9) is deduced he relaionships insananeous sress-srain concerning boh models which give he nonlineariy in he elasic par of he herediary equaion (). In he case of incompressible maerials beween he insananeous Young modules one has he following relaion E E G (10) (1 ) This is because he Poisson raio is 0.5. Moreover, from equaions (7), (8) here is he relaionship Q( ) Q( ) ( ). (11) S( ) So Then from equaions (9), (10) and (11) i is obained he following relaions sress-elongaion E 1 ( ) (1a) E 1 1 ( ) ( ( 1)) (1b) Equaions (1) may also be represened as a relaion sress-srain according o eq. (5a). These expressions have he form E (1 ) 1 ( ) (1a) 1 E (1 ) 1 1 ( ) ( ( 1)) (1b) 1 (1 ) Equaions (1) should be resolved abou he srains in order o obain he non-linear sress-srain relaion for highly deformable maerials. These equaions are of hird and fourh degree respecively and can be represened in he following form using equaions (1) 1 0, (14a) E 60
K. B. Hadjov e al.: Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions 4 ( ) 0.5 ( ) 0.5 0 (14b) E The soluion of equaion (14a) using he Cardano s formula [10] aking ino accoun (5a) looks like ( ) 0.5 0.5 ( / E) 0.5 0.5 ( / E) 1 (15а) The soluion of equaion (14b) using he Ferrari s formula [10] aking ino accoun (5a) looks like 0.5 W( ) ( ( ) y( ) ( )/ W( )) ( ) 1, 4( ) (15b) where: (0.5 ) / E ( ), 8( ) (0.5 ) / E ( ) 1 8( ) 4( ) 4 (0.5 ) (0.5 ) / E ( ) 4 56( ) 16( ) 8( ) 1 1 1 Q ( ) ( ) ( ) ( ) ( ), 108 8 1 P ( ) ( ) ( ) 1 1 Q ( ) P ( ) R( ) Q( ), 4 7 5 P( ) y( ) ( ) R( ), 6 R( ) W ( ) ( ) y( ). Nonlinear elasoviscosiy a large deformaions The relaionship of he Cauchy s deformaions wih he sresses in he form (15) is used as insananeous nonlineariy in he herediary equaion (). Equaion () based on (15) looks like [ ] n ( ) o( ) 1 ( / o) K( τ) dτ (16) The creep law (16) conains parameers ha need o be experimenally idenified. The necessary experimenaions are as follows: - Insananeous (wih consan srain rae or oherwise) idenified from (15). Noe ha eq. (15a) is a very successful equaion conaining only one parameer o describe he complex nonlinear elasiciy and his is he elasic module E. - Tess based on he curves of sress relaxaion (or creep) in he linear region a small imposed srains. 0 These parameers are idenifiable from () a low sresses 0. - Tess based on he isochrone curves of creep (or relaxaion). These parameers are idenifiable from equaion () a high sresses > 0. EXPERIMENTAL Fig.1 and Fig. show he insananeous nonlineariy of he buadieneniril (BN) and polyizoprene (PI) rubbers. Fig.1. Relaive ensile force - relaive elongaion according o he Neo-Hookean and according o he Mooney-Rivlin law for boh rubbers Boh rubbers are produced on open mixer wih dimensions 400x150 [mm] a T=50 0 C. Time of vulcanizaion is deermined on oscillaing rheomeer ype Moncano. Vulcanizaion of esed specimens is done a hydraulic press wih auomaic conrol of pressure and emperaure. Fig.. Tensile sress - srain according o he Neo- Hookean - equaion (14a) and o he Mooney-Rivlin law - equaion (14 b) for boh rubbers. Solid lines - he Neo- Hookean, dashed lines - he M. Rivlin model, above - PI rubber, below - BN rubber 61
K. B. Hadjov e al.: Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions Fig. and Fig.4 illusrae he idenificaion of he above parameers concerning boh rubbers. Figure shows he sress relaxaion of he BN by consan srain. Fig.4 shows he corresponding creep curve for BN a sresses on he limi of he nonlineariy (see Table 1a and Table 1b). The same is done in Fig.5 and Fig.6 for he PI rubber. 0.5 Relaxaion curve for BN rubber sress in [MPa] 0.7 0.65 0.6 Relaxaion curve - PI rubber sress in [MPa] 0.48 0.46 0.44 0.4 0.55 0 100 00 00 ime in [h] Fig.5. Sress relaxaion of PI rubber Creep curve - PI rubber 0.4 0 50 100 150 00 50 00 50 0.5 ime in [h] Fig.. Sress relaxaion of BN rubber srain 0.45 0. Creep curve for BN rubber 0.4 0 100 00 00 srain 0.19 0.18 0.17 ime in [h] Fig.6. Creep curve of PI rubber In he nex wo Fig.7 and Fig.8 are shown he nonlinear isochrone curves sress-srains a sresses greaer han he limi of non-lineariy. 0.16 0.15 0 50 100 150 00 50 00 50 ime in [h] Fig.4. Creep curve of BN rubber Noe: In Fig, and Fig.4 are shown all he experimenal poins. In he nex figures - jus he averaged values. Creep curves are no necessary o predic he behaviour and can be used only for verificaion. In pracice, he parameers in he Kolunov s kernel can be idenified from he relaxaion curves, which can be obained much easily han he creep curves. Fig.7. Rubber nonlineariy of PI rubber (isochrones a 0 and 100 hours) Fig.9 and Fig.10 show he creep of he PI and BN rubbers a wo sress levels according o equaion (16). 6
K. B. Hadjov e al.: Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions From boh figures we can see ha under bigger sress for sudied ime inerval creep has unidenified characer. Finally, in Table 1(a and b) are sysemaized all he parameers obained from he experimenal curves above. Table 1. (a and b) Elasoviscous characerisics of he invesigaed rubbers Table 1a. Insananeous (elasic) characerisics Elasoviscous Characer insananeous (elasic) Fig.8. Rubber nonlineariy of BN rubber (isochrones a 10, 50 and 100 hours) Parameers Dimension PI rubber BN rubber Е o N χ Ϛ [MPa] [MPa] - - - 1.80 0.75 1.71 0.14 0.64.05 0.50 1.55 0.018 0.960 Table 1b. Herediary (viscous) characerisics elasoviscous Characer herediary (viscous) Parameers A α β Dimension - - - PI rubber 0.00 00.97 0.0140 BN rubber 0.009 00.77 0.0089 RESULTS AND DISCUSSION Fig.9. Creep a large deformaion of PI Fig.10. Creep a large deformaion of BN For he BN rubber boh models give good resuls. Therefore for BN rubbers i is used he neo- Hookean model. I has he advanage ha i does no require he enire force-displacemen curve o deermine he parameers (he elasic module is sufficien). The PI rubber however, requires he use of he -parameer model of Mooney-Rivlin. Sress-srain curves according o equaions (1a, b) or also o (15a, b) appear as shown in Fig.. The experimenal resuls in figure 1 show ha he neo- Hookean law well describes he insananeous behaviour of he BN rubber bu o he PI rubber should be applied he more flexible model of Mooney-Rivlin. The assumpion of incompressibiliy for he rubbers is perfecly accepable. For our maerials he Poisson s raio values are 0.485 o he BN rubber and 0.49 o he PI rubber. 6
K. B. Hadjov e al.: Analyic nonlinear elaso-viscosiy of wo ypes of BN and PI rubbers a large deformaions The herediary heory wih kernel of Kolunov well describes he emporary effecs due o he viscosiy. CONCLUSION Using he inegral equaions of Volerra o describe he nonlinear elasoviscous behaviour are obained equaions predicing he ime dependen behaviour aking ino accoun he sress-srain nonlineariy. On he basis of he Neo-Hookean and Mooney-Rivlin models concerning he insananeous nonlineariy a large deformaions are derived he respecive srain-sress consiuive relaions in analyical form. Boh Neo-Hookean and Mooney-Rivlin models well describe he insananeous mechanical behaviour a large deformaions for he BN rubber. Concerning he PI rubber, only he Mooney-Rivlin model is able o well predic he sress-srain insananeous consiuive relaion. These srain-sress consiuive relaions are incorporaed in he herediary heory of Volerra o obain he complex ime dependen mechanical behaviour of rubbers a large deformaions. The heoreical predicions show very good coincidence wih he experimenal daa for PI and BN rubbers. REFERENCES 1 N. R. Legge, G. Holden, H. E. Schroeders, Thermoplasic elasomers: A comprehensive review, Hanser, Munich, (1987). S. Ilisch, H. Menge, H-J. Radusch, Kauschuk und Gummi Kunssoffe, 5, 06-1, (000). K. Naskar, Rubber Chemisry and Technology, 80, 504 510, (007). 4 A. Alexandrov, Z. Tzolov, 6 h European Polymer Federaion Symposium on Polymeric Maerials, Thesaloniki, Creece, 79-8, (1996). 5 A. Alexandrov, Z. Tzolov, Inernaional Journal of Differenial Equaions and Applicaions, (4), 59-6, (001). 6 N. Rabonov, Elemens of he herediary mechanics of solids, Moskow, Nauka, 1977. 7 M. A. Kolunov, Creep and relaxaion, Moskow, Nauka, (1976). 8 A. Alexandrov, Z. Tzolov, Inernaional Journal of Differenial Equaions and Applicaions, 5 (), 147-154, (00). 9 J. Salençon, Mécanique du Coninu, Tomes 1,,, Ellipses, (1995). 10 M. Abramowiz, I. A. Segun, (Eds.). Handbook of Mahemaical Funcions wih Formulas, Graphs and Mahemaical Tables, 9h prining. New York, Dover, (197). 64